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Aloiza [94]
3 years ago
15

A room has eight switches, each of which controls a different light. Initially, exactly five of the lights are on. Three people

enter the room, one after the other. Each person independently flips one switch at random and then exits the room. What is the probability that after the third person has exited the room, exactly six of the lights are on? Express your answer as a common fraction.
Mathematics
1 answer:
aivan3 [116]3 years ago
8 0

Answer:

The probability that 3 lights are on after the third person exited the room is 39/128

Step-by-step explanation:

In order for 6 switches to be on at the end, we need exactly 2 people turning on a switch and the other one turning one off. There are 3 possibilities:

  • The first two persons turn the switch in and the last one turns it off
  • The first and last person turn the switch in and the middle one tunrns it off
  • The first person turns the switch off and the 2 remaining turn the switch in

Note that after turning off one switch one more switch will be available to be switched in and one less will be available to be switch off. The contrary happens when someone turns in a switch.

Lets calculate the probability for the first scenario. The probability for the first person to turn the switch in is 3/8, because there are 3 lights off. For the second person, there will be only 2 lights off, thus, the probability for him or her to turn the switch in is only 2/8, leaving only 1 light off and 7 on. The third person will have, as a consecuence, a probability of 7/8 to turn off one of the 7 switches. This gives us a probability of 3/8 * 2/8 * 7/8 = 21/256 for the first scenario.

For the second scenario we will have a probability of 3/8 for the first person, a probability of 6/8 for the second one (he has to turn a switch off this time), and a probability, again, of 3/8 for the third one, giving us a probability of 3/8*6/8*3/8 = 27/256 for the second scenario.

For the third scenario, the first person has to turn off the switch, and it has a probability of 5/8 of doing so. The second person will have 4 switches to turn on, so it has a probability of 4/8 = 1/2, and the third person will have one switch less, thus, a probability of 3/8 of turning a switch on. Therefore, the probability of the third scenario is 5/8*1/2*3/8 = 15/128 = 30/256

By summing all the three disjoint scenarios, the probability that six lights are on is 21/256+27/256+30/256 = 78/256 = 39/128.

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Let theta be an angle in quadrant 2 such that cos theta =-3/4. Find the exact values of csc theta and cot theta.
Anna35 [415]

Step-by-step explanation:

Since cos(\theta) = -\frac{3}{4} we can get some information from this. First of all cos(\theta) is defined as \frac{adjacent}{hypotenuse}. So the adjacent side is 3 and the hypotenuse is 4. Using this we can find the opposite side to find sin(\theta) to calculate csc and cot of theta. So using the Pythagorean Theorem we can solve for the missing side. Also I forget to mention we have to calculate the sign of the adjacent side and the hypotenuse. Since you're given that the angle is in quadrant 2, that means the x-value is going to be negative, and the y-value is going to be negative. And the x really represents the adjacent side and the y represents the opposite. So the adjacent side is what's negative. and the opposite is positive

Pythagorean Theorem:

a^2+b^2=c^2

(-3)^2 + b^2 = 4^2

9 + b^2 = 16

b^2 = 7

b = \sqrt{7}

So now we can calculate sin(\theta).

sin(\theta) = \frac{\sqrt{7}}{4}

Now to calculate the exact value of csc you simply take the inverse. This gives you csc(\theta)=\frac{4}{\sqrt{7}}. Multiplying both sides by sqrt(7) to rational the denominator gives you \frac{4\sqrt7}{7}.

Now to calculate cot(theta) you  find the inverse of tan. Tan is defined as tan(\theta) = \frac{sin(\theta)}{cos(\theta)}. So all you do is take the inverse which is cot(\theta) = \frac{cos(\theta)}{sin(\theta)}.

Plug values in

cot(\theta) = \frac{-\frac{3}{4}}{\frac{4\sqrt{7}}{7}}

Keep, change, flip

-\frac{3}{4} * \frac{7}{4\sqrt7}

Multiply:

-\frac{21}{16\sqrt7}

Multiply both sides by sqrt(7)

-\frac{21\sqrt7}{112}

This is the value of cot(theta)

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